Was Fermat too bold about her last theorem?

TenaliRaman

Was Fermat too bold about her last theorem?

i never knew Fermat was female :D

-- AI

TenaliRaman

|_/\/\40!!

:d

-- Ai

mathwonk

A remark, possibly of interest, involving confusing terminology. Wiles' background was not strictly in math of ellipses, but in math of elliptic curves. These curves may have arisen of the study of ellipses, in the time of the Bernoulli's say, in the following way, thus possibly giving rise to their name.

I.e. an ellipse is a non singular plane curve of degree 2, but an elliptic curve is a non singular plane curve of degree 3, or a curve "birationally equaivalent" to such a cubic.

Now as you know, if you have taught calculus and arc length, it is not so easy to compute the arclength of a parabola, and an ellipse is even harder, i.e. virtually impossible in elementary terms.

The integral you get when you try an ellipse if I recall is something like 1/sqrt(1+x^4). Now the plane curve defined by the reciprocal of this integral, y^2 = 1+x^4, is a singular plane quartic (the singularity is at infinity in the projective plane), but still birationally equivalent to a non singular plane cubic, hence is an "elliptic" curve.

So elliptic curves are one level more difficult than ellipses, which is actually quite a lot. And an elliptic curve is not an ellipse but the curve arising from the arclength integral for an ellipse. This beautiful story is told wonderfully well by C. L. Siegel, (using the lemniscate as his example, instead of an ellipse), in volume one of his Topics in complex function theory.

Zurtex

Sorry mathwonk, I knew that I just mis-typed.

mathwonk

Sorry to assume you did not know that.

I was a little too glib in my response though. I was not actually able to verify my claim that the arclength of an ellipse leads to the integral I asserted it did. It seems that is easier to check for the arclength of the lemniscate, which is probbaly why Siegel chose that one, or maybe for historical reasons, as he attributes the discovery to a Count Fagnano of Italy.
When i reviewed my work from calc last year, I found that I merely found a statement in my calculus book that asserted that the arclength integral for the ellipse reduced to an integral they called an "elliptic integral". so I am taking my calculus book's word for it that the arclength of an ellipse leads to an elliptic integral but I did not see how it worked out that way myself.

ryokan

Gauss wrote: "Ich gestehe zwar, dass das Fermatsche Theorem als isolierter Satz fuer mich wenig Interesse hat..." (I confess that the Fermat theorem holds little interest for me as an isolated result...)
It was in a letter to Heinrich Olbers (May 1816)

mathwonk

I regret Gauss felt that way. Others with a longer time line, have bequeathed us the enormous heritage of knowledge, such as the theory of factorization, unique and otherwise by Kummer et al, growing out of their work on Fermat's problem, not to mention the big boost given to analytic number theory by Wiles' ideas.

So it would have been nice if Gauss had actually stated some of his problems for the rest of humanity to work on after his lifetime.

ryokan

No. He was a man with moustache.
Obviously, I made a mistake, when I wrote "her" to talk about his last theorem :blush: